2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Algebra D1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yuichiro Taguchi
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 5-6 Thu
- Class
- -
- Course Code
- MTH.A408
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 4Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
In number theory, L-functions are associated with various arithmetic objects. In this course, we explain the interpretation of L-functions based on Galois representations and discuss important results/conjectures on them. This course together with "Advanced Course in Algebra C1" given in the 3Q forms one set of contents. Advanced topics (Dwork theory, modularity) are dealt with in the latter half "D1".
Course description and aims
Students are expected to:
- understand the proof of analyticity of the L-functions of φ-modules by Dwork theory.
- understand the notion of modularity of an L-function, and the proof of the modularity in the 1-dimensional case.
Keywords
L-function, Galois representation, modularity, φ-module, Dwork theory
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | L-functions of φ-sheaves (Part 1) | Details will be provided during each class session |
| Class 2 | L-functions of φ-sheaves (Part 2) | Details will be provided during each class session |
| Class 3 | L-functions of φ-sheaves (Part 3) | Details will be provided during each class session |
| Class 4 | Algebraic Hecke characters | Details will be provided during each class session |
| Class 5 | L-functions of CM Abelian varieties (Part 1) | Details will be provided during each class session |
| Class 6 | L-functions of CM Abelian varieties (Part 2) | Details will be provided during each class session |
| Class 7 | L-functions of CM Abelian varieties (Part 3) | Details will be provided during each class session |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to indulge themselves in L-functions and Galois representations.
Textbook(s)
None required.
Reference books, course materials, etc.
None required.
Evaluation methods and criteria
Course scores are evaluated by homework assignments (100%). Details will be announced during the course.
Related courses
- MTH.A407 ： Advanced topics in Algebra C1
- ZUA.A334 ： Advanced courses in Algebra C1
- ZUA.A334 ： Advanced courses in Algebra D1
Prerequisites
Basic knowledge in undergraduate algebra
Other
None in particular.